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Theorem 0nnq 10335
Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
0nnq ¬ ∅ ∈ Q

Proof of Theorem 0nnq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0nelxp 5566 . 2 ¬ ∅ ∈ (N × N)
2 df-nq 10323 . . . 4 Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd𝑥) <N (2nd𝑦))}
32ssrab3 4032 . . 3 Q ⊆ (N × N)
43sseli 3938 . 2 (∅ ∈ Q → ∅ ∈ (N × N))
51, 4mto 200 1 ¬ ∅ ∈ Q
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2114  wral 3130  c0 4265   class class class wbr 5042   × cxp 5530  cfv 6334  2nd c2nd 7674  Ncnpi 10255   <N clti 10258   ~Q ceq 10262  Qcnq 10263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-opab 5105  df-xp 5538  df-nq 10323
This theorem is referenced by:  adderpq  10367  mulerpq  10368  addassnq  10369  mulassnq  10370  distrnq  10372  recmulnq  10375  recclnq  10377  ltanq  10382  ltmnq  10383  ltexnq  10386  nsmallnq  10388  ltbtwnnq  10389  ltrnq  10390  prlem934  10444  ltaddpr  10445  ltexprlem2  10448  ltexprlem3  10449  ltexprlem4  10450  ltexprlem6  10452  ltexprlem7  10453  prlem936  10458  reclem2pr  10459
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